Global number field 32.32.71166659227026865952165739269552697151542080277732014965436610606901057.1

• Label: 32.32.7116665922...1057.1
• Degree: $32$
• Signature: $[32, 0]$
• Discriminant: $193^{31}$
• Root discriminant: $163.73$
• Ramified prime: $193$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{32}$ (as 32T33)

Normalized defining polynomial

\( x^{32} - x^{31} - 93 x^{30} + 210 x^{29} + 3538 x^{28} - 12479 x^{27} - 65768 x^{26} + 350581 x^{25} + 485064 x^{24} - 5339623 x^{23} + 2808135 x^{22} + 44434017 x^{21} - 84476060 x^{20} - 170361963 x^{19} + 681732194 x^{18} - 77972047 x^{17} - 2529499095 x^{16} + 3133974961 x^{15} + 3455889877 x^{14} - 10578598738 x^{13} + 3732760107 x^{12} + 12702259726 x^{11} - 15151938680 x^{10} - 638400055 x^{9} + 11610249674 x^{8} - 6566502528 x^{7} - 1290174274 x^{6} + 2262714296 x^{5} - 382096988 x^{4} - 214409519 x^{3} + 59339724 x^{2} + 2830728 x + 14969 \)

Invariants

Degree:		$32$
Signature:		$[32, 0]$
Discriminant:		\(71166659227026865952165739269552697151542080277732014965436610606901057=193^{31}\)
Root discriminant:		$163.73$
Ramified primes:		$193$
This field is Galois and abelian over $\Q$.
Conductor:		\(193\)
Dirichlet character group:		$\lbrace$$\chi_{193}(1,·)$, $\chi_{193}(3,·)$, $\chi_{193}(129,·)$, $\chi_{193}(8,·)$, $\chi_{193}(9,·)$, $\chi_{193}(151,·)$, $\chi_{193}(14,·)$, $\chi_{193}(143,·)$, $\chi_{193}(150,·)$, $\chi_{193}(23,·)$, $\chi_{193}(24,·)$, $\chi_{193}(27,·)$, $\chi_{193}(166,·)$, $\chi_{193}(169,·)$, $\chi_{193}(42,·)$, $\chi_{193}(43,·)$, $\chi_{193}(50,·)$, $\chi_{193}(179,·)$, $\chi_{193}(184,·)$, $\chi_{193}(185,·)$, $\chi_{193}(190,·)$, $\chi_{193}(192,·)$, $\chi_{193}(67,·)$, $\chi_{193}(69,·)$, $\chi_{193}(72,·)$, $\chi_{193}(64,·)$, $\chi_{193}(81,·)$, $\chi_{193}(112,·)$, $\chi_{193}(121,·)$, $\chi_{193}(124,·)$, $\chi_{193}(170,·)$, $\chi_{193}(126,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{109} a^{27} - \frac{27}{109} a^{26} + \frac{6}{109} a^{25} + \frac{29}{109} a^{24} - \frac{18}{109} a^{23} - \frac{7}{109} a^{22} + \frac{30}{109} a^{21} + \frac{7}{109} a^{20} + \frac{47}{109} a^{19} + \frac{40}{109} a^{18} - \frac{27}{109} a^{17} - \frac{44}{109} a^{16} + \frac{20}{109} a^{15} - \frac{21}{109} a^{14} - \frac{38}{109} a^{13} - \frac{15}{109} a^{12} + \frac{54}{109} a^{11} + \frac{39}{109} a^{10} - \frac{42}{109} a^{9} + \frac{16}{109} a^{8} - \frac{51}{109} a^{7} - \frac{16}{109} a^{6} - \frac{44}{109} a^{5} - \frac{42}{109} a^{4} + \frac{4}{109} a^{3} + \frac{44}{109} a^{2} - \frac{46}{109} a - \frac{8}{109}$, $\frac{1}{109} a^{28} + \frac{40}{109} a^{26} - \frac{27}{109} a^{25} + \frac{2}{109} a^{24} + \frac{52}{109} a^{23} - \frac{50}{109} a^{22} + \frac{54}{109} a^{21} + \frac{18}{109} a^{20} + \frac{1}{109} a^{19} - \frac{37}{109} a^{18} - \frac{10}{109} a^{17} + \frac{31}{109} a^{16} - \frac{26}{109} a^{15} + \frac{49}{109} a^{14} + \frac{49}{109} a^{13} - \frac{24}{109} a^{12} - \frac{29}{109} a^{11} + \frac{30}{109} a^{10} - \frac{28}{109} a^{9} + \frac{54}{109} a^{8} + \frac{24}{109} a^{7} - \frac{40}{109} a^{6} - \frac{31}{109} a^{5} - \frac{40}{109} a^{4} + \frac{43}{109} a^{3} + \frac{52}{109} a^{2} - \frac{51}{109} a + \frac{2}{109}$, $\frac{1}{109} a^{29} - \frac{37}{109} a^{26} - \frac{20}{109} a^{25} - \frac{18}{109} a^{24} + \frac{16}{109} a^{23} + \frac{7}{109} a^{22} + \frac{17}{109} a^{21} + \frac{48}{109} a^{20} + \frac{45}{109} a^{19} + \frac{25}{109} a^{18} + \frac{21}{109} a^{17} - \frac{10}{109} a^{16} + \frac{12}{109} a^{15} + \frac{17}{109} a^{14} - \frac{30}{109} a^{13} + \frac{26}{109} a^{12} + \frac{50}{109} a^{11} + \frac{47}{109} a^{10} - \frac{10}{109} a^{9} + \frac{38}{109} a^{8} + \frac{38}{109} a^{7} - \frac{45}{109} a^{6} - \frac{24}{109} a^{5} - \frac{21}{109} a^{4} + \frac{1}{109} a^{3} + \frac{42}{109} a^{2} - \frac{11}{109} a - \frac{7}{109}$, $\frac{1}{30193} a^{30} - \frac{21}{30193} a^{29} + \frac{119}{30193} a^{28} - \frac{18}{30193} a^{27} - \frac{5896}{30193} a^{26} + \frac{5587}{30193} a^{25} - \frac{2632}{30193} a^{24} + \frac{11185}{30193} a^{23} - \frac{112}{277} a^{22} - \frac{14132}{30193} a^{21} + \frac{6108}{30193} a^{20} - \frac{13751}{30193} a^{19} - \frac{13957}{30193} a^{18} + \frac{14196}{30193} a^{17} + \frac{14520}{30193} a^{16} + \frac{8387}{30193} a^{15} - \frac{5637}{30193} a^{14} - \frac{11021}{30193} a^{13} + \frac{5628}{30193} a^{12} + \frac{3657}{30193} a^{11} + \frac{13887}{30193} a^{10} + \frac{12904}{30193} a^{9} + \frac{7169}{30193} a^{8} + \frac{10636}{30193} a^{7} + \frac{3487}{30193} a^{6} + \frac{7294}{30193} a^{5} - \frac{11329}{30193} a^{4} + \frac{9574}{30193} a^{3} + \frac{11690}{30193} a^{2} - \frac{14894}{30193} a + \frac{11678}{30193}$, $\frac{1}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{31} + \frac{1396507123818223485209754402329439067571042059663111756146946635332911}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{30} + \frac{306310575189403577215638401094667138072900124115649890840056800156724311}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{29} + \frac{937615056053044181704337557844810989936278489147982403224139671524560}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{28} - \frac{277503096565812892451483024868815493287589885354617601562814781990442912}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{27} + \frac{495594538348463567524052942159185498973226661658161242305486030082952023}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{26} - \frac{29239664252816441021953066167529958674637868800558515142151816100082615774}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{25} + \frac{19272825817189826453104395060327010271851321459367789626269072518190755083}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{24} + \frac{36936754452736965692326848953529318101567517306829608849727557149272150089}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{23} - \frac{38226414025146792408209283112905659797781915964845149794642157930447569318}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{22} - \frac{29992727041974110857442449995723569153249503684502041461517380398734290207}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{21} + \frac{21366213418584076348994202334501387281711043406097336247423776768360160761}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{20} + \frac{6396695472118213122902085878678077058490782344460138652872845153673183806}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{19} - \frac{44755070334935059328611731475611000290664693868147552308895447499243151546}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{18} + \frac{33577852353806166158093536678365248712954497374651669728258948639964971064}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{17} + \frac{14294923369624059370519409177188334632886660499528876373949366638391151457}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{16} + \frac{1493869439959016648845230743977181039916633605135492806121679455320784794}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{15} - \frac{38440885912489752852739586248175533954378040660639438385055119939832054672}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{14} - \frac{8951851251347891676319497780097711487620665729348289021720769878815970781}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{13} - \frac{49685347508415004267472004530813198157508038664639180471881409588657509663}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{12} - \frac{29738303163575678515496353395222995388326315270131652787270538451685611420}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{11} - \frac{16507731964222916824203652210206811201658084033346134938552192538287676055}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{10} + \frac{48188617193349507673316467673641346628173199072753213345100185906536131207}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{9} - \frac{23969766421996308066887458613173998663698757862377092053682136955024895717}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{8} + \frac{49864615462722117696242498717064851454694162570302354556374807762265383202}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{7} + \frac{11011875594612784040558455558604228357922187917154816994113017126236334625}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{6} + \frac{7712480199740460000580286338234456345343292417201590092711384054070504791}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{5} - \frac{14272376344200405912600186440282109343018183866375338551202783773219976668}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{4} + \frac{25718962045426335401757227091941179054489361256365263344696356046912415033}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{3} - \frac{38912817047077705191860450839538794731106976687506448327192392904989014194}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{2} - \frac{746296672922043619428248489309353485053485191226337700772741401983713917}{100877246011414201550061355185574630947470476963762602792140977676019421393} a - \frac{194799629934785461542633515391544786164568851525441569973020697696165}{6739077160225412622757789777912661563729739926766156910424275347452697}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$31$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 17081301615656049000000000 \) (assuming GRH)

Galois group

$C_{32}$ (as 32T33):

A cyclic group of order 32

The 32 conjugacy class representatives for $C_{32}$

Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{193}) \), 4.4.7189057.1, 8.8.9974730326005057.1, 16.16.19202582299769315484813817587637057.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	$16^{2}$	${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$	$32$	${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$	$32$	$32$	$32$	$32$	$16^{2}$	$32$	$16^{2}$	$32$	$32$	${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$	$32$	$32$	${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
193	Data not computed